Let $R$ be a ring (not necessarily with identity). Is there a name for an element $a\in R$ for which $ab=0$ for all $b\in R$? Similarly, for an element $a\in R$ for which $ba=0$ for all $b\in R$? Note, a unital ring has exactly one of each, just zero in both cases, as if $ab=0$ […]

# Author: Q+A Expert

Show that in general there is no rigid reference frame defined in any q+a.

## Logic uniqueness and injection

Definition of uniqueness is $\exists y [P(y) \land \forall x[P(x) \rightarrow x=y]]$. Then, $\exists y [P(y) \land \forall x[\lnot P(x) \lor x=y]]$. Distributing $P(y)$, we have $\exists y \forall x[P(y) \land \lnot P(x) \lor P(y) \land (x=y)]$. So, we have $\exists y \forall x[P(y) \subseteq P(x) \rightarrow P(y) \land (x=y)]$, then $\exists y \forall x[P(y) […]

$X$ and $Y$ are two independent random variables distribute Uniform$[0,1]$, where $M = \min(X,Y)$, $N = \max(X,Y)$. I want to find the density of the couple $(M,N)$. We can start by finding the marginal distributions, doing so gives $$f_M(m) = 2(1-m),~~ m \in [0,1]$$ $$f_N(n) = 2n,~~ n \in [0,1] $$ I found this answer […]

Given a continuous function $f : [0,1) \rightarrow \mathbb{R}$, $f(0) = 0$, $f(x) >0$ for $x>0$ and $$ \lim_{x \rightarrow 1} \frac{1}{f(x)} = 0.$$ Prove that for every $c>0$ there exists an $a \in [0,1)$ with $f(a) > c$, also prove that $f([0,1)) = [0, \infty )$. Since $ \lim_{x \rightarrow 1} \frac{1}{f(x)} = 0$ […]

## density function and measure null

If $P = fd\mu $, $Q = gd\mu$ and $P = h dQ$ do we have $$ \mu(g=0)=0 $$ My goal is to show that $h = \frac{f}{g} $ almost $\mu$ everywhere. thanks and regards.

## it’s no use doing sth

Thank you for clicking into this page to help me. In the grammar book, I have learned that a gerund can be preceded by a pronoun (usually more acceptable, a possessive pronoun) to show who it is that does this action. And regarding gerund, there is a structure ‘it’s no use doing sth’ , where […]

The following heuristic argument for the prime number theorem was taken from https://sites.williams.edu/Morgan/2008/10/11/heuristic-derivation-of-prime-number-theorem/. Frank Morgan attributes it to Hugh Bray via Greg Martin. Suppose that there is a nice probability function $P(x)$ that a large integer $x$ is prime. As $x$ increases by $\Delta x = 1$, the new potential divisor $x$ is prime with […]

The maximum weight that an elevator in an apartment complex can accommodate is 800kg. The average adult weight be about 70 kgs with a variance of 200. What is the probability that the lift safely reaches the ground when there are 10 adults in the lift?

I know you have to do around the same thing as the 9 by 16 rectangle to a 12 by 12, but I need to know how to make my 9 by 4 into a 6 by 6 with 3 pieces. There are 2 ways. I need the other. If you can, can you please […]

I have been stuck on this problem for some time and I am not sure how to approach it: “How many positive integers less than 100,000 have digits containing 4,5,6 in that particular order?” I am thinking that I would multiply 10X10X3X2X1 since the two of the five digit spaces can hold any digit between […]